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Entropy 2018, 20(9), 690; https://doi.org/10.3390/e20090690

Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms

1
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 62529, Saudi Arabia
2
Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 12 August 2018 / Revised: 5 September 2018 / Accepted: 7 September 2018 / Published: 11 September 2018
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Abstract

In this paper, we obtain the upper bounds for the normalized δ -Casorati curvatures and generalized normalized δ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be η -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations. View Full-Text
Keywords: Casorati curvatures; statistical manifolds; Sasaki-like statistical space forms; η-Einstein Casorati curvatures; statistical manifolds; Sasaki-like statistical space forms; η-Einstein
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Alkhaldi, A.H.; Aquib, M.; Siddiqui, A.N.; Shahid, M.H. Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms. Entropy 2018, 20, 690.

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